Generating State Variable Description

[hadron23]

Hello,

I came across a problem in some literature and was curious about how to solve it,

Given a system described by,

$$\ddot{y} + 2\dot{y} - 3y = \dot{u} - u$$

Convert the above into state-space form with input $$\dot{u}$$ and output y.

Define the state vector and determine the matrices A(t),B(t),C(t),D(t) such that,

$$\dot{x} = A(t)x + B(t)\dot{u}$$
$$y = C(t)x + D(t)\dot{u}$$

Any ideas?

 

[sandhi]

Can anybody tell me what will be the state space equation for the circuit in the below link. I will be very happy to know this.http://i783.photobucket.com/albums/yy113/sandhi_prashant/Statespaceequation.jpg

with regards,
Sandhi

 

[oswaler]

I would approach this problem by first understanding the concept of state-space representation. This is a mathematical tool commonly used in control theory to describe the behavior of a dynamic system over time. It involves representing the system using a set of state variables, which are a minimum number of variables that can fully describe the system's behavior at any given time.

In this case, the system is described by a second-order differential equation with an input and output. To convert it into state-space form, we first need to define our state vector. In this case, it would be:

x = [y, \dot{y}]^T

Next, we can use this state vector to determine the matrices A(t), B(t), C(t), and D(t) using the following equations:

\dot{x} = A(t)x + B(t)u
y = C(t)x + D(t)u

Here, u represents the input, which in this case is \dot{u}. By substituting the given system equation into these equations, we can solve for the matrices A(t), B(t), C(t), and D(t). Once we have these matrices, we can use them to analyze the behavior of the system and design control strategies if needed.

In summary, the process of converting a system into state-space form involves defining state variables and using equations to determine the matrices that fully describe the system's behavior. I hope this helps in solving your problem.

 

[oswaler]

Hello,

The first step in solving this problem would be to generate the state variable description of the given system. This involves defining the state vector and determining the matrices A(t), B(t), C(t), and D(t). The state vector, denoted by x, is a set of variables that fully describes the state of the system at any given time t. In this case, we can define the state vector as x = [y, \dot{y}]^T, where y is the output and \dot{y} is the first derivative of y.

Next, we can rewrite the given system in state-space form as follows:

\dot{x} = A(t)x + B(t)\dot{u}
y = C(t)x + D(t)\dot{u}

Where A(t) is a 2x2 matrix, B(t) is a 2x1 matrix, C(t) is a 1x2 matrix, and D(t) is a 1x1 matrix. These matrices can be determined by rearranging the given system into the standard form of state-space equations:

\dot{x} = Ax + Bu
y = Cx + Du

Therefore, the matrices can be determined as follows:

A = \begin{bmatrix} 0 & 1 \\ -3 & 2 \end{bmatrix}
B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
C = \begin{bmatrix} 1 & 0 \end{bmatrix}
D = \begin{bmatrix} 0 \end{bmatrix}

In summary, to solve the given problem, we need to generate the state variable description and determine the corresponding matrices A(t), B(t), C(t), and D(t). These matrices will then be used to solve the system using standard state-space techniques. I hope this helps. Let me know if you have any further questions.